Subject: Calculus (Mathematics)
Topic: Maxima-Minima (New Lesson)
The students shall be able to:
1. State the meaning of maxima-minima.
2. Solve and locate the point where the maximum and minimum points of a
given curve appears using the First Derivative Test (FDT).
Step 1- Students are given a written pre-test to assess their general knowledge of functions that includes graphs and evaluating zeros, critical values, minima and maxima to assess students’ readiness for mastering the concepts and skills necessary for this lesson.
Step 2- The results of the pre-test will determine how the instruction will be planned.
a - The students whose assessment results indicate that they lack the pre-requisite skills, then these students will be my second level group and they will be grouped together for reviewing the required knowledge base of understanding and mastery of prerequisite skills of the lesson.
b- The students with special needs will constitute the third level of differentiating the instructional plan. These are the students whose test results indicate that they have mastered the concepts or the students who require accommodations. For those who have mastered the basic skills and concepts, they will be assigned enrichment activities such applications, in depth study, and computer based modeling. For students with special needs, one-on-one pull out tutoring sessions will be necessary initially to review the prerequisites necessary and introduce the new concepts. This part will be conducted by an outside resource.
c- For the majority of the class the following activity will introduce the lesson’s concepts:
Students are placed in groups of three with similar learning style and mixed abilities. Each group is given a foot long piece of a wire, masking tape and a foot ruler. Students were asked to bend the wire to form a multi- peak arc. Students were then asked to use the masking tape and tape the wire in the note book such that at least one end point of the wire coincides with the longer side of a page in their note book and the entire wire is contained within that page. Students are asked to shape wire such that it represents the graph of what type of general relationship. Written instruction is provided that introduces the necessary vocabulary by drawing language structures from students via leading questions such as what would a concise scientific name would be for the peaks and the valleys, the highest peak lowest peak and the lowest valley.
(1) The students will refine their definitions and share them with the class, until a mathematically acceptable definition has been arrived at.
(2) The instruction then asks the student to draw a tangent line at the peaks and the valleys and explore the value of the slope of these tangent lines and report what they have in common. Hopefully the students will realize that all these tangent lines will have a slope of zero.
(3) The instruction queries the students about the relationship between the slope of a line and the derivative. Then use that information to hypothesize about the derivative’s value at the maxima and the minima (peaks & valleys). Hopefully the students will arrive at the hypothesis that the derivative’s value at the maxima and the minima is zero.
(4) Using leading questions in the form of Q & A, the groups are instructed to relate the sign of the slope of an increasing function, i.e., rising tangent lines, and decreasing functions, falling tangent lines. Then the groups are instructed to relate this observation to the derivative of the segment of the wire on either side of the maxima and the minima. The students hopefully will recognize that the signs are opposite.
The evaluation for level 3 groups will be based upon the activities assigned. They will be assessed on the concepts in the context of assigned tasks.
Level 1 group will be assessed as follows:
(1) The teacher then asks the students to use the information obtained in step (4) to write out the wording of a hypothesis that pertains to the signs of the derivative of the function and the maxima/minima. The students will refine their definitions and share them with the class. This process will be repeated until a mathematically acceptable definition of the FTD has been arrived at.
(2) The teacher then asks the classroom, given the rule of a function, how would you use calculus to arrive at the max and min points of that function? Hopefully the students will respond by saying, “Take the derivative of the function, set it to equal zero, and solve.”
(3) The teacher asks, “How do you know what you have obtained in step (2) is a minimum or maximum point of the function?” and hopefully the students will respond, “by the order of the signs of the derivatives in either side of the maximum and minimum points. A (+, 0,-) indicates a maximum point, while a (-,0,+) indicates a maximum points.
You have seen that if a point is maximum or minimum, then the derivative is zero. Is the converse true? Justify your answer.